| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2017 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Conic sections |
| Type | Parabola normal intersection problems |
| Difficulty | Hard +2.3 This is a challenging Further Maths question requiring multiple sophisticated techniques: implicit differentiation for the normal equation, understanding parabola focus properties, solving a system where a circle touches a parabola (requiring the normal to pass through the circle's center), and finding the area using the tangency condition. The multi-step reasoning connecting parts (a)-(d) and the geometric insight needed for part (c) place this well above average difficulty. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07m Tangents and normals: gradient and equations |
8. The parabola $C$ has equation $y ^ { 2 } = 4 a x$, where $a$ is a positive constant.
The point $P \left( a t ^ { 2 } , 2 a t \right)$ lies on $C$.
\begin{enumerate}[label=(\alph*)]
\item Using calculus, show that the normal to $C$ at $P$ has equation
$$y + t x = a t ^ { 3 } + 2 a t$$
The point $S$ is the focus of the parabola $C$.\\
The point $B$ lies on the positive $x$-axis and $O B = 5 O S$, where $O$ is the origin.
\item Write down, in terms of $a$, the coordinates of the point $B$.
A circle has centre $B$ and touches the parabola $C$ at two distinct points $Q$ and $R$.
Given that $t \neq 0$,
\item find the coordinates of the points $Q$ and $R$.
\item Hence find, in terms of $a$, the area of triangle $B Q R$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2017 Q8 [12]}}